Ridge regression with adaptive additive rectangles and other piecewise functional templates
This work addresses the challenge of nonconvex knot placement in piecewise functional regression for statisticians and data analysts, offering an incremental improvement through a reduced-variable optimization scheme.
The authors tackled the problem of functional linear regression by proposing an L2-penalized algorithm that shrinks the coefficient function towards a data-driven piecewise template, specifically a sum of adaptively positioned rectangles, to improve predictive power and interpretability.
We propose an $L_{2}$-based penalization algorithm for functional linear regression models, where the coefficient function is shrunk towards a data-driven shape template $γ$, which is constrained to belong to a class of piecewise functions by restricting its basis expansion. In particular, we focus on the case where $γ$ can be expressed as a sum of $q$ rectangles that are adaptively positioned with respect to the regression error. As the problem of finding the optimal knot placement of a piecewise function is nonconvex, the proposed parametrization allows to reduce the number of variables in the global optimization scheme, resulting in a fitting algorithm that alternates between approximating a suitable template and solving a convex ridge-like problem. The predictive power and interpretability of our method is shown on multiple simulations and two real world case studies.